Problem 86 - Algebra 1: Concepts and Skills

Question

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. (Lesson 9.2) $$ 9+x^{2}=49 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \sqrt{40}\) and \(x = -\sqrt{40}\).

Step 1: Isolate the variable term

To isolate the variable term \(x^2\), subtract 9 from both sides of the equation. This gives \(x^2 = 49 - 9\), which simplifies into \(x^2 = 40\).

Step 2: Apply square root

To solve for \(x\), take the square root of both sides of the equation. Remember that when you take the square root of both sides of an equation, you need to consider both the positive and negative roots. Therefore, \(x = \sqrt{40}\) and \(x = -\sqrt{40}\).

Step 3: Simplify the result

Since \(\sqrt{40}\) doesn't result to a clean integer result, keep it as is. So, the solutions for the equation are \(x = \sqrt{40}\) and \(x = -\sqrt{40}\).

Key Concepts

Integer Solutions

Solving a quadratic equation often results in solutions that show up as integers. An integer solution refers to a solution that is a whole number, without any fractions or decimals. In the equation \(9 + x^2 = 49\), we first isolate the term with the variable \(x^2\). Once we have \(x^2 = 40\), we then apply the square root to both sides. Ideally, we'd find that \(x\) is an integer. However, in this case, 40 is not a perfect square.

A perfect square is an integer that is the square of another integer. For example, 4, 9, and 16 are perfect squares. Numbers that aren't perfect squares, like 40, result in radical expressions instead of clean integer solutions when square rooted. Always double-check if the number on the opposite side of the variable term is a perfect square to determine if integer solutions are possible.

Square Root Method

The square root method is a valuable technique for solving quadratic equations. This method is very effective when you can isolate the variable term squared, like \(x^2\), on one side of the equation. For instance, in our equation \(x^2 = 40\), once we have eliminated the constant, we can apply the square root method.

Here's how it works:

• Isolate the squared variable (e.g., \(x^2\)).
• Take the square root of both sides of the equation.
• Remember, every positive square root also has a negative counterpart.

This means when taking the square root, you need to consider both the positive and negative possibilities (\(x = \sqrt{40}\) and \(x = -\sqrt{40}\)). The square root method is often straightforward, but it's essential to remember to consider both positive and negative roots as they represent different, legitimate solutions to the equation.

Radical Expressions

Radical expressions occur when you can't simplify the square root into a whole number. In our example, the expression \(\sqrt{40}\) is such a case because 40 doesn't have a perfect square root. A radical expression involves the use of the square root symbol and can often be simplified to some extent.

Simplifying \(\sqrt{40}\) involves factoring out perfect squares. Here's a quick breakdown:

• Factor 40 into prime components: \(40 = 2^3 \times 5\).
• Identify perfect square factors: \(\sqrt{4 \times 10}\).
• Simplify the radical: \(\sqrt{4}\) simplifies to 2, giving \(2\sqrt{10}\).

Thus, the solutions are more neatly expressed as \(2\sqrt{10}\) and \(-2\sqrt{10}\). Radical expressions can be simplified by finding any factor within them that is a perfect square, and this simplification provides a clearer and often more practical form for further calculations.